The classical discrete dynamics between \(N\)  spherically symmetrical objects
with masses \( m^N=m_1, m_2,..,m_N\)  and positions \(\textbf {r}^N(t)=\textbf {r}_1(t), \textbf {r}_2(t),..,\textbf {r}_N(t)\)  is obtained
by Eq. (A2) and Eq. (A3). The momentum and angular momentum for a conservative system of the \(N\)  objects is conserved due to Newton's third law, and
the energy is also conserved [1]}, [2]}, [3]}, so Newton's time reversible and symplectic discrete dynamics has the same invariances as his analytic dynamics.
Newton's discrete algorithm make it possible to simulate systems of interacting objects for billions of time steps without \(any\)  adjustments for the infinite
range of the gravitational forces or corrections for symplecticity,
time reversibility, conservation of momentum, angular momentum and energy.
